LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-24

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Rational arithmetic is well defined

Statement

Sum, product, and negation of Arithmetic on the rationals are independent of the chosen representatives.

Facts & Assumptions

Given: Pairs with $(a, b) \sim (a^{'}, b^{'})$ and $(c, d) \sim (c^{'}, d^{'})$ (The rationals as equivalence classes of pairs of integers), all second components nonzero.

[L1]

$Z$ is a commutative ring (The integers form a commutative ring).

Proof

technique · direct

1.1given

By hypothesis $a b^{'} = a^{'} b$ and $c d^{'} = c^{'} d$ .

2.1step 1.1L1

Product: $(a c) (b^{'} d^{'}) = (a b^{'}) (c d^{'}) = (a^{'} b) (c^{'} d) = (a^{'} c^{'}) (b d)$ , so $(a c, b d) \sim (a^{'} c^{'}, b^{'} d^{'})$ .

2.2step 1.1L1

Sum: $(a d + c b) (b^{'} d^{'}) = (a b^{'}) (d d^{'}) + (c d^{'}) (b b^{'}) = (a^{'} b) (d d^{'}) + (c^{'} d) (b b^{'}) = (a^{'} d^{'} + c^{'} b^{'}) (b d)$ , so $(a d + c b, b d) \sim (a^{'} d^{'} + c^{'} b^{'}, b^{'} d^{'})$ .

2.3step 1.1L1

Negation: from $a b^{'} = a^{'} b$ , $(- a) b^{'} = (- a^{'}) b$ , so $(- a, b) \sim (- a^{'}, b^{'})$ .

Rational arithmetic is well defined

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources